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# Central Limit Theorem

The central limit theorem (CLT) is one of the most useful theorems in statistics. It states that regardless of the population distribution, as long as all the variables are independently and identically distributed (i.i.d.), the sampling distribution of means will approach a normal distribution with mean equal to the population mean as the sampling size increases.

Consider the population of Canada and imagine that you would like to find out what the mean height is. You start by taking a sample of size N and take the mean of that sample. But, this does not guarantee at all, that this will be the same as the population's mean. Now imagine that you continue to take more random samples of size N and find their means. Plotting all those means will give you a sampling distribution of means. The CLT states that this sampling distribution will approach a normal distribution as N grows large. Furthermore the mean of this sampling distribution will be approximately equal to the population mean as well. The sampling distribution variance will be approximately the population variance divided by N. The sampling standard deviation will be the square root of that. The central limit theorem evidently gives the ability to solve for the population mean with more accuracy than using one sample.

### Probability Example: High school graduates

High school graduates: The National Center for Educational Statistics reported that 82% of freshmen entering public high schools in the U.S. in 2009 graduated with their class in 2013. A random sample of 135 freshmen is chosen. Find the probability that less than 80% of freshmen in the sample graduated. Find the probability th

### Statistical Control

A product development group determines that it must have a fiber, which among other properties, as a minimum tensile strength of 2.500 gm in 99 percent of the fiber used. Manufacturer-ABC offers to supply such fiber and a contract is arranged. Manufacturer-ABC knows that the standard deviation () for the process of 0.020 gm.

### Normal Distribution on a Portfolio

1) The daily returns on a portfolio are normally distributed with a mean of 0.001 and a standard deviation of 0.002. What is the probability that the average return for the portfolio over the next 100 days exceeds 0.0015? 2) In May 1983, after an extensive investigation by the Consumer Product Safety Commission, Honeywell

### Normal Approximation to Binomial Distribution - Weld Example

Smith is a weld inspector at a ship yard. He knows from keeping track of good and substandard welds that 5% will be substandard. If he checks 300 of 7500 welds, what is the probability that he will find less than 20 substandard welds? To solve the problems on the up-coming test we will use either the "normalcdf" or InvNorm" f

### Statistical Analysis: Are we smarter than our parents

Read the article entitled, "Are We Smarter than Our Parents?" in chapter 5 of your textbook. This article addresses a study by Dr. James Flynn of the rise of the IQ rate over generations, and how statistics are involved in tracking this phenomenon, especially with reference to the material in chapter 5. Write a paper in which

### Central Limit Theorem and Probabilities Under Standard Normal Curve

Will you check my work on 1, 2 & 5 and then help me with 3, 4 & 6... I don't know if the formula is still the same for all of them or different. 1. The heights of eighteen-year-old men are approximately normally distributed with a mean (µ) of 68 inches and a standard deviation (σ) of 3 inches. What is the probability that

### Normal Probabilities and Confidence Intervals

I would like to have the attached sample questions completed in excel format, so that I can see the formulas used. Thanks

### Central Limit Theorem Explained in Layman's Terms

Could you please explain Central Limit Theorem to me in layman's terms and then read the article linked below and detail how it was used therein? Thank you. http://www.tandfonline.com/doi/abs/10.1080/02664761003692308?journalCode=cjas20#.UdSZ1fnVCuk

### The solution gives detailed steps on determining type I and type II errors in a specific hypothesis testing and using confidence interval method to make the conclusion of the same test. Next, central limit theorem is well defined and explained in details.

(b) The quality control engineer for a furniture manufacturer is interested in the mean amount of force necessary to produce cracks in stressed oak furniture. She performs a two-tailed test of the null hypothesis that the mean for the stressed oak furniture is 650. What are Type I and Type II errors for this problem? (c) A 95

### Application of Probability in Duracell Battery Life

1. Duracell Ltd claim that the life of their batteries in motorized soft toys is approximately normally distributed with a mean of 102.9 hours and a standard deviation of 16.5 hours. The best 65% of batteries would last beyond how many hours? Answer correct to 2 decimal places. 2. The average and standard deviation of the amo

### Insurance Agents Nationwide in Ohio

Then the 27 listed insurance agents Nationwide Insurance in the metropolitan area of Toledo, Ohio. Calculate the average number of years that have worked in Nationwide. See attached. a) Select a random sample of four agents. If the Random numbers selected are: 02,59,51,25,14,29,77,69 and 18, then which dealers will be incl

### Statistics Control Limits

A tire company is interested in monitoring the process that produced tread thickness on its tires. Every hour 4 tires are selected from production and the tread thickness is measured. Data for the past 25 days is shown as follows: (Please see the attached file.) a. What type of control chart would you recommend be used i

### Center Line and Control Limits

500 units, randomly chosen from an assembly line, are inspected each day and the number of defective are recorded. 22 33 24 20 18 24 24 29 18 27 31 46 31 24 22 a. Find the center line and the control limits for this process. b. Does the process appear to be in control? Why?

### Central Limit Theorem and the Poisson Distribution

A business has 5 telephone lines. The first rings every minute on the average, the second every ½ minute, the third every 2 minutes, the fourth every 5 minutes, and the fifth every 1/10 minute, on the average. Let Xt = #(calls in [0,t]). (a) What is the exact distribution of Xt? Find (b) E(X60) and (c) P(X60 < 850) using

### Normal Distribution and Central Limit Theorem

The weight of 3rd grade children is normally distributed with a mean of 64 and a standard deviation of 10. Two children are selected at random from a very large population. (a.) What is the probability that both will weigh more than 70 pounds? (b.) What is the probability that one of the children will weigh more than 65 poun

### Central limit theorem..

The average balance for customer accounts in The Reserve Fund at the time it was frozen by the Securities Exchange Commission (SEC) was \$22,500, with a standard deviation of \$7,500. The SEC overseers want to draw a sample of 100 accounts to help assess the impact of the fund's freeze on the account holders. Precisely (that is, u

### Central Limit Theorem Conversation

Please see attachment. And please provide clear and thorough information so that I can understand it in future problems. Please provide assistance in the following 10 parts: 1. Boss has a question about a population: 2. To costly or can't do population so forced to sample 3. With sampling, will be sampling error (CLT

### Probability

In the Department of Education at UR University, student records suggest that the population of students spends an average of 5.5 hrs per week playing organized sports. The population's standard deviation is 2.2 hrs per week. Based on a sample of 121 students, healthy lifestyles Inc. (HLI) would like to apply the central limit t

### Normal Approximation to Binomial and Central Limit Theorem

Problems are also attached. 1. A fair die is rolled 25 times. Let X be the number of times a six is obtained. Find the exact value of P( X=6) and compare it with a normal approximation of P( X=6). 2. (Dice Sums). Suppose a fair die is rolled 1000 times. Compute an approximation to the probability that the sum of the 1000 r

### Central Limit Theorem: Experiment for number of rolls of dice for normal distribution

Visit the following Web site Central Limit Theorem Applet and read what is posted: http://www.stat.sc.edu/~west/javahtml/CLT.html You will choose from the pull down menu at the bottom of the page both the number of dice and the number of rolls at a time. When you "click" you will be virtually rolling your dice. Complete th

### Descriptive Statistics: Mean for E1 following a standard distribution

See attached file and please solve calculations in red. 2 Using the descriptive statistics data determined during Week One's weekly problem discussion, the mean for EI followed a standard distribution with a mean of 132.83 and a standard deviation of 15.68. If we selected another random sample of 50 participant

### Sampling methods and Central Limit Theorem for Nationwide Insurance Agents in Toledo, OH

Listed below are the 27 Nationwide Insurance agents in the Toledo, Ohio, metropolitan area. We would like to estimate the mean number of years employed with Nationwide... a. We want to select a random sample of four agents. The random numbers are ... b. Use the table of random numbers to select your own sample of four agent

### Statistics: 15 Problems

1. A population is normally distributed, with a mean of 23.45 and a standard deviation of 3.8. What is the probability of each of the following? a. Taking a sample of size 10 and obtaining a sample mean of 22 or more b. Taking a sample of size 4 and getting a sample mean of more than 26 2. The Statistical Abstract of the

### Statistics: Normal Distribution, probabilities, values, central limit theorem

Included with each section or problem are reference examples and end of section exercises that can be used as a guide. Be sure to show your work in case partial credit is awarded. To receive full credit, work must be shown if applicable. Section 5.1: Introduction to Normal Distribution and the Standard Normal Distribution 1.

### Statistics: 8.46, 8.62, Central Limit Theorem, Population shape of concern

8.46 A random sample of 10 miniature Tootsie Rolls was taken from a bag. Each piece was weighed on a very accurate scale. The results in grams were 3.087 3.131 3.241 3.241 3.270 3.353 3.400 3.411 3.437 3.477 (a) Construct a 90 percent confidence interval for the true mean weight. (b) What sample size would be necessary

### Central limit

A worldwide organization of academics claims that the mean IQ score of its members is 118, with a standard deviation of 15. A randomly selected group of 40 members of this organization is tested, and the results reveal that the mean IQ score in this sample is 116.8. If the organization's claim is correct, what is the probability

### Central Limit Theorem and Population Shape

1. State the main points of the Central Limit Theorem for a mean. 2. Why is population shape of concern when estimating a mean? What does sample size have to do with all?

### Distribution characteristics foundations

The normal distribution has several characteristics that make it the underlying foundation of much of inferential statistical tools. List these characteristics and explain how they contribute to the power of the distribution as foundation of inferential decision making.

### Statistics discussion: Control chart distributions differences for a large sample

Please help with the following problem. Provide at least 200 words in the solution. Control charts are a popular (and often reasonable) way to statistically monitor a processes' performance. They are used for both variables (i.e. quantitative measures) and attributes (qualitative measures). Our authors state that the underly

### Central limit theorem for population

Sample sizes of 50 are selected from a normal population with a mean of 78.1 and standard deviation of 16.2. Calculate the probability that a single value selected at random will be between 71.1 and 85.1